Your question more has to do with using the Amplify effect than about using the ReplayGain plugin.

Bottom line if you don't want to read all the stuff below: Don't allow clipping, and ignore that second box with a number on it.

Details:

As a general rule, never allow clipping. Clipping is when your wave forms hit the limits - top or bottom of the recording window. You end up losing data from the wave form that cannot be recovered, and this can cause distortion.

In your case in the video, you were making the file quieter (hence the minus sign when you go to amplify), so you're making the wave forms smaller. In this case you'll never run into a problem with clipping.

However, if your file were too quiet, you might have a case where it won't let you amplify as much as ReplayGain tells you to. This is because you have a spike (or more than one) in your recording that will hit the limit (it would clip) if it were amplified that much. You could either zoom into the spike and select just the spike to amplify it by a negative number to squish it down, or you could run the Compression effect on your whole file.

After compression, run ReplayGain again and amplify by how much it tells you.

This is for files that are too quiet, which isn't your case in the video.

---

The other box with the number in it, "new peak amplitude", is not something to worry about. Highlight your recording, go to Effects/amplify, and change the first box to 0. If your highest spike is 2.5 dB below the maximum, it'll show in there as -2.5 dB.

When you go to Effects/Amplify, it will default the first box to the highest you can amplify before hitting maximum (clipping), and the second box will show 0. Whatever number you put in the first amplify box will be added or subtracted from this maximum.

There are proper peak meters which are available as free downloads which will run alongside Audacity and give clear indications of peaks being hit and also give an update of true maximum peaks as well as being able to be set for gain and reference levels. One such download is (a VST) called dpMeter2 (by TBProAudio). Search and it will come up with a short manual as well.

The meters in Reaper will give a max retained reading on peaks as well.

Zero DBFS (decibels at full scale) is at the point of going over into distortion. In proper terminology you should keep to about -10 DBFS which will give you a good recording.

There can be as problem with peaks (spikes) which depend on mic technique to avoid them. But they often do happen even with the best mic technique. If there are only a few (as it should be hopefully) then you can reduce just the peaks by a few Db by isolation and reduction. You can do this even in audacity.

From there, leave all defaults and just click on OK. You should get a response such as:
ReplayGain level: -1.8 dB or
ReplayGain level: 2.7 dB.

This means that you have to use Effect > Amplify and set your amplification to -1.8 or 2.7 dB, respectively.Keep the defaults, make sure to not check the box for "Allow clipping", and confirm by clicking OK.

(I rearranged the last sentence so the "click OK" is after keeping the defaults.)

Stay away from the latest release of Audacity. They are still trying to work out the problems with their upgrade. I use version 2.1.3 and if you can record with your input level between a -12 and -18dbs, the rest is a piece of cake. I did a video on this. You can view it here. You can scrub to 6 min 22 sec to skip the intro.